The Order Product Prime Probability and Commutativity Degree for Some Finite Groups

Statistical approach often provides techniques in exploring more on the algebraic properties of finite groups. In this paper, two new group probabilities are defined in which the group elements having certain common characteristics are considered. Let G be a finite group, the order product prime probability is defined as the probability that two randomly selected elements x and y in G satisfy |x||y|=ps and the order product prime commutativity degree is the probability that elements satisfying the above property commutes. Some formulas for computing these probabilities for dihedral groups are obtained.